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What is exactly the domain of validity of the identity:

Sum[1/(k + j - 1), {j, 1, i}] == HarmonicNumber[k - 1 + i] - HarmonicNumber[k - 1]  

when k is a real number?

More generally, could I have a reference about the way HarmonicNumber[x] is calculated when x is not an Integer?

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As indicated here, HarmonicNumber[x] is the same as EulerGamma + PolyGamma[x+1]. So the more detailed remarks of PolyGamma would apply. For a more mathematical explanation see also this article. In the end it comes down to the gamma function which is defined as an integral.

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